Patterns in random walks and Brownian motion CSP conference in honor of J. Pitman 20 − 21 June 2014, UC San Diego Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman 20 June, 2014 Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion
Scenario # 1 Question Given some distribution of a process X with continuous paths, is there a random time T such that ( B T + u − B T ; 0 ≤ u ≤ 1 ) has the same distribution as ( X u , 0 ≤ u ≤ 1 ) ?. Examples : Brownian/pseudo bridge, Brownian meander, normalized excursion, Bessel ( 3 ) , Vervaat bridges . . . etc. The question here has some affinity to the well-known Skorokhod embedding problem . The question is related to splitting theorems of post- T Markov processes. Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion
Scenario # 2 Question Given a Borel measurable subset S ⊂ C [ 0 , 1 ] , can we find a random time T such that ( B T + u − B T ; 0 ≤ u ≤ 1 ) ∈ S with probability one ? Examples : E := { w ∈ C [ 0 , 1 ]; w ( t ) > w ( 1 ) = 0 for 0 < t < 1 } ; M := { w ∈ C [ 0 , 1 ]; w ( t ) > 0 for 0 < t ≤ 1 } ; BR λ := { w ∈ C [ 0 , 1 ]; w ( 1 ) = λ } ; FP λ := { w ∈ C [ 0 , 1 ]; w ( t ) > w ( 1 ) = λ for 0 ≤ t < 1 } ; VB λ := { w ∈ FP λ ; ζ := inf { t > 0 ; w ( t ) < 0 } > 0 } . . . etc . Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion
Scenario # 3 Question Given for each n a collection A n of patterns of length n, what is the order of the expected waiting time E T ( A n ) until one of the elements of A n is observed in a random walk ? Examples : E 2 n := { w ∈ SW ( 2 n ); w ( i ) > 0 for 1 ≤ i ≤ 2 n − 1 and w ( 2 n ) = 0 } ; M 2 n + 1 := { w ∈ SW ( 2 n + 1 ); w ( i ) > 0 for 1 ≤ i ≤ 2 n + 1 } ; √ BR λ, n := { w ∈ SW ( n ); w ( n ) = λ n } where λ n ∼ λ n ; FP λ, n := { w ∈ SW ( n ); w ( i ) > w ( n ) = λ n for 0 ≤ i ≤ n − 1 } . . . etc . Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion
Response to Scenario # 3 Theorem There exists C E > 0 such that 1 3 E T ( E 2 n ) ∼ C E n 2 ; There exists C M > 0 such that 2 E T ( M 2 n + 1 ) ∼ C M n ; There exists C λ BR > 0 such that 3 E T ( BR λ, n ) ∼ C λ BR n ; There exists c λ FP and C λ FP > 0 such that 4 5 c λ FP n ≤ E T ( FP λ, n ) ≤ C λ 4 . FP n Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion
Response to Scenario # 2 Theorem a.s. ∄ random time T such that ( B T + u − B T ; 0 ≤ u ≤ 1 ) ∈ 1 E := { w ∈ C [ 0 , 1 ]; w ( t ) > w ( 1 ) = 0 for 0 < t < 1 } ; a.s. ∄ random time T such that ( B T + u − B T ; 0 ≤ u ≤ 1 ) ∈ 2 RBR := { w ∈ C [ 0 , 1 ]; w ( t ) ≥ w ( 1 ) = 0 for 0 ≤ t ≤ 1 } ; For each λ < 0 , a.s. ∄ T s.t. ( B T + u − B T ; 0 ≤ u ≤ 1 ) ∈ 3 VB λ := { w ∈ FP λ ; ζ := inf { t > 0 ; w ( t ) < 0 } > 0 } , where FP λ := { w ∈ C [ 0 , 1 ]; w ( t ) > w ( 1 ) = λ for 0 ≤ t < 1 } . Consequence : no normalized excursion , reflected bridges , Vervaat bridges in a Brownian path ! Idea : Williams’ path decompositions , or fragmentation argument . Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion
Response to Scenario # 1 Theorem For each of the following three processes X := ( X u , ≤ u ≤ 1 ) there is some random time T such that ( B T + u − B T ; 0 ≤ u ≤ 1 ) has the same distribution as X : the meander X = ( m u ; 0 ≤ u ≤ 1 ) ; 1 the co-meander X = ( � m u ; 0 ≤ u ≤ 1 ) ; 2 the Bessel ( 3 ) process X = ( R u ; 0 ≤ u ≤ 1 ) . 3 Idea : Brownian meander by Itˆ o’s excursion theory and the other two by acceptance-rejection method . Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion
Our favorite open problem Open problem Can we find a random time T such that ( B T + u − B T ; 0 ≤ u ≤ 1 ) has the same distribution as Brownian bridge ( b 0 u ; 0 ≤ u ≤ 1 ) ? The bridge pattern BR 0 is achieved by the bridge-like process ( B T + u − B T ; 0 ≤ u ≤ 1 ) , where T := inf { t > 0 ; B t − B t + 1 = 0 } . The bridge-like process can be inferred from the work of ⇒ P ( T > t ) is computed. Slepian and Shepp = From simulation, the above bridge-like process is not Brownian bridge. The related Slepian zero set has rich properties, work in progress with Jim Pitman. Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion
Remerciement Thank you for your attention, AND Happy birthday, Jim ! Wenpin Tang Statistics Department, UC Berkeley Joint work with Jim Pitman Patterns in random walks and Brownian motion
Recommend
More recommend
